Question

In Exercises 6.139 to \(6.142,\) use the normal distribution to find a confidence interval for a difference in proportions \(p_{1}-p_{2}\) given the relevant sample results. Give the best estimate for \(p_{1}-p_{2},\) the margin of error, and the confidence interval. Assume the results come from random samples. A \(95 \%\) confidence interval for \(p_{1}-p_{2}\) given that \(\hat{p}_{1}=0.72\) with \(n_{1}=500\) and \(\hat{p}_{2}=0.68\) with \(n_{2}=300 .\)

Short answer

The best estimate for the difference in proportions is 0.04, the standard error is the obtained SE value, and the margin of error is calculated as the z-value (1.96 for 95% confidence) times the standard error. The confidence interval for a difference in proportions is calculated as \(0.04 \pm ME\).

Step-by-step solution

Calculating Sample Proportions

First, you need to calculate the sample proportions \(\hat{p}_{1}\) and \(\hat{p}_{2}\) if not given. They can be calculated as the number of success out of the total number of trials. For this question we have, \(\hat{p}_{1}=0.72\) and \(\hat{p}_{2}=0.68\) which means that out of all trials 72% and 68% were successful respectively.

Calculating Best Estimate for Difference in Proportions

The best estimate for the difference in proportions \(p_{1}-p_{2}\) is the difference of the observed/sampled proportions \(\hat{p}_{1}-\hat{p}_{2}\). For this question, the best estimate for the difference in proportions would be calculated as \(0.72 - 0.68 = 0.04\)

Calculate the Standard Error

The standard error of the difference in two proportions is calculated using the formula:\( SE = \sqrt{\hat{p}_{1} (1-\hat{p}_{1})/n_{1}+\hat{p}_{2} (1-\hat{p}_{2})/n_{2}}.\) In this case, substituting the relevant values we get:\(SE = \sqrt{0.72*(1-0.72)/500 + 0.68*(1-0.68)/300}\)

Calculating Margin of Error

The margin of error for the difference of proportions can be calculated as \(\text{ME} = z * SE,\) where \(z\) is the z-value corresponding to the given level of confidence (for 95%, it is approximately 1.96). \(\text{So ME} = 1.96 * SE\)

Calculating Confidence Interval

Finally, the confidence interval for the difference of sample proportions can be calculated using the formula: \(\hat{p}_{1} - \hat{p}_{2} \pm z * SE\). So the confidence interval = \(0.04 \pm ME\)